Critical points of the optimal quantum control landscape: a propagator approach

Numerical and experimental realizations of quantum control are closely connected to the properties of the mapping from the control to the unitary propagator. For bilinear quantum control problems, no general results are available to fully determine when this mapping is singular or not. In this paper we give suffcient conditions, in terms of elements of the evolution semigroup, for a trajectory to be non-singular. We identify two lists of "way-points" that, when reached, ensure the non-singularity of the control trajectory. It is found that under appropriate hypotheses one of those lists does not depend on the values of the coupling operator matrix

Singularity-free quantum tracking control of molecular rotor orientation

Quantum tracking control aims to identify applied fields to steer the expectation values of particular observables along desired paths in time. The associated temporal fields can be identified by inverting the underlying dynamical equations for the observables. However, fields found in this manner are often plagued by undesirable singularities. In this paper we consider a planar molecular rotor, and derive singularity-free tracking expressions for the fields that steer the expectation of the orientation of the rotor along desired trajectories in time. Simulations are presented that utilize two orthogonal control electric fields to drive the orientation of the rotor along a series of designated tracks

Characterization of the Critical Sets of Quantum Unitary Control Landscapes

This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schr\"odinger equation. We examine the critical point structure of the kinematic landscapes J_F (U) = ||(U-W)A||^2 and J_P (U) = ||A||^4 - |Tr(AA'W'U)|^2 defined on the unitary group U(H) of a finite-dimensional Hilbert space H. The parameter operator A in B(H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on a subspace of state space, namely the column space of A. The analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse-Bott functions on U(H). Landscapes based on the intrinsic (geodesic) distance on U(H) and the projective unitary group PU(H) are also considered. These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes.Comment: 15 pages, 3 figure

Exploring Quantum Control Landscape Structure

A common goal of quantum control is to maximize a physical observable through the application of a tailored field. The observable value as a function of the field constitutes a quantum control landscape. Previous works have shown, under specified conditions, that the quantum control landscape should be free of suboptimal critical points. This favorable landscape topology is one factor contributing to the efficiency of climbing the landscape. An additional, complementary factor is the landscape \textit{structure}, which constitutes all non-topological features. If the landscape's structure is too complex, then climbs may be forced to take inefficient convoluted routes to finding optimal controls. This paper provides a foundation for understanding control landscape structure by examining the linearity of gradient-based optimization trajectories through the space of control fields. For this assessment, a metric $R\geq 1$ is defined as the ratio of the path length of the optimization trajectory to the Euclidean distance between the initial control field and the resultant optimal control field that takes an observable from the bottom to the top of the landscape. Computational analyses for simple model quantum systems are performed to ascertain the relative abundance of nearly straight control trajectories encountered when optimizing a state-to-state transition probability. The collected results indicate that quantum control landscapes have very simple structural features. The favorable topology and the complementary simple structure of the control landscape provide a basis for understanding the generally observed ease of optimizing a state-to-state transition probability.Comment: 27 pages, 7 figure

Collision kernel and interatomic potential

This is the published version, also available here: http://dx.doi.org/10.1103/PhysRevA.33.3067.We present a detailed study of the influence of the form and strength of the interatomic potential on the one-dimensional elastic collision kernel W(v’z,vz), a quantity of interest in the study of the effects of velocity-changing collisions on laser spectroscopic line shapes. We find that the absolute magnitudes of collision kernels are very sensitive while normalized collision kernels are moderately sensitive to the potential form used. This indicates the importance of employing realistic interatomic potentials and reliable differential cross sections in the accurate determination of collision kernels. For the case of the Lennard-Jones (12,6) potential, we found a universal semiclassical Lennard-Jones (SCLJ) analytical model function, which is the combination of a semiclassical expression for small to medium scattering angles and a classical expression for large scattering angles, capable of providing correct average quantum-mechanical behaviors of differential cross sections for all scattering angles. This greatly facilitates the (often time-consuming) numerical evaluation of the collision kernel integrals and exhibits the correct collision kernel line profiles. It is found that the SCLJ collision kernel consists of a strongly peaked forward diffractive zone (small-angle scatterings), reflecting the nature of velocity resonance, as well as a broad wing region due to large-angle scatterings. Ambiguities associated with the drawbacks of the hard-sphere model, the small-angle classical long-range model, and the classical Lennard-Jones model are analyzed and clarified. While our analysis is confined to the Na-Ar and Ar-Ar systems, the conclusions derived from this study are general and are expected to be also applicable to other systems where both the long- and short-range interactions play essential roles in velocity-changing collisions

Semiclassical many-mode Floquet theory. III. SU(3) dynamical evolution of three-level systems in intense bichromatic fields

This is the publishers version, also available here: http://dx.doi.org/10.1103/PhysRevA.31.659.The SU(3) dynamical evolution of three-level systems at two-photon resonance induced by two strong linearly polarized monochromatic fields is studied exactly by means of the semiclassical many-mode Floquet theory (MMFT) recently developed by the authors. Within the rotating-wave approximation (RWA), Hioe and Eberly have recently shown that the eight-dimensional SU(3) coherent vector S→ characterizing the time evolution of three-level systems can be factored into three independent vectors of dimensions three, four, and one, at appropriate two-photon resonance conditions. In practice, however, if the laser-atom interactions occur away from the two-photon resonance, or if the RWA is not valid, etc., this Gell-Mann–type SU(3) dynamical symmetry will be broken. It is shown in this paper that instead of solving the time-dependent generalized Bloch equations, the SU(3) dynamical evolution of the coherent vector S→ as well as various symmetry-breaking effects can be expediently studied by the use of the MMFT and expressed in terms of a few time-independent quasienergy eigenvalues and eigenvectors. Furthermore, we have extended the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent Floquet Hamiltonian into a 3×3 effective Hamiltonian, from which useful analytical properties of the SU(3) coherent vector can be easily obtained. The combination of the MMFT and the GVV method thus greatly facilitates the study of the dynamical evolution. Pictorial comparison of the exact and the RWA results of the time evolution of the eight-dimensional coherent vector under several different physical conditions is presented and discussed at length

Semiclassical many-mode Floquet theory. IV. Coherent population trapping and SU(3) dynamical evolution of dissipative three-level systems in intense bichromatic fields

This is the published version, also available here: http://dx.doi.org/10.1103/PhysRevA.32.377.The many-mode Floquet theory (MMFT) recently developed by authors is extended to incorporate the irreversible damping mechanisms for the nonperturbative treatment of the dynamical evolution of dissipative three-level systems at two-photon or multiphoton coherent resonance trapping conditions induced by two strong linearly polarized monochromatic fields. It has been recently shown by several workers that under the rotating-wave approximation (RWA), population may be permanently trapped in the three-level system if the coherent monochromatic fields are exactly two-photon resonant with the initial and final states, decoupled from the intermediate decaying level. In practice, the inclusion of the non-RWA terms necessarily modifies the resonant trapping conditions and behavior. In this paper we extend the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the non-Hermitian two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent non-Hermitian Floquet Hamiltonian to a 3×3 effective Hamiltonian, from which essential properties of the coherent population-trapping behavior as well as the dynamical evolution of the dissipative SU(3) coherence vector S→(t) can be readily obtained and expressed in terms of only three complex quasienergy eigenvalues and eigenvectors. The MMFT-GVV studies show that the RWA two-photon resonant trapping condition is substantially modified by the effects of non-RWA terms, and that the system can be ‘‘quasitrapped’’ for only a finite amount of time characterized by a small imaginary energy (width) associated with a coherent superposition state of the initial and final levels. Furthermore, it is found that the initially eight-dimensional coherence vector S→(t) evolves predominantly to a one-dimensional scalar at the two-photon or multiphoton resonant quasitrapping conditions. Detailed results and pictorial representations of the population trapping and SU(3) dissipative dynamical evolution are presented

Nonadiabatic approach for resonant molecular multiphoton absorption processes in intense infrared laser fields

This is the published version, also available here: http://dx.doi.org/10.1063/1.445612.A nonperturbative approach for efficient and accurate treatment of the molecular multiphoton absorption (MPA) quantum dynamics in intense infrared (IR) laser fields is presented. The approach is based on the adiabatic separation of the fast vibrational motion from the slow rotational motion, incorporating the fact that the IR laser frequency is close to the frequencies of adjacent vibrational transitions. One thus first solves the quasivibrational energy (QVE) states (or, equivalently, the dressed vibrational states) with molecular orientation fixed. This reduces the computationally often formidable (vibrational‐rotational) Floquet matrix analysis to a manageable scale, and, in addition, provides useful physical insights for understanding the nonlinear MPA dynamics. The QVE levels are found to be grouped into distinct energy bands, characterized by the IR frequency, with each band providing an effective potential for molecular rotation. Whereas the interband couplings are totally negligible, the intraband nonadiabatic angular couplings are the main driving mechanisms for inducing resonant vibrational–rotational multiphoton transitions. The utility of the method is illustrated by a detailed study of the sequential MPA spectra for 12C 16O molecule, including state‐to‐state multiquantum transitions and transitions from initially thermally distributed states as a whole. Results are presented for the case of IR laser intensity 50 GW/cm2 and frequencies ranging from 2115 to 2165 cm− 1. Excellent agreement of the MPA spectra obtained by the nonadiabatic approach and the exact Floquet matrix method was observed in all fine details

Exploration of the memory effect on the photon-assisted tunneling via a single quantum dot: A generalized Floquet theoretical approach

The generalized Floquet approach is developed to study memory effect on electron transport phenomena through a periodically driven single quantum dot in an electrode-multi-level dot-electrode nanoscale quantum device. The memory effect is treated using a multi-function Lorentzian spectral density (LSD) model that mimics the spectral density of each electrode in terms of multiple Lorentzian functions. For the symmetric single-function LSD model involving a single-level dot, the underlying single-particle propagator is shown to be related to a 2 x 2 effective time-dependent Hamiltonian that includes both the periodic external field and the electrode memory effect. By invoking the generalized Van Vleck (GVV) nearly degenerate perturbation theory, an analytical Tien-Gordon-like expression is derived for arbitrary order multi- photon resonance d.c. tunneling current. Numerically converged simulations and the GVV analytical results are in good agreement, revealing the origin of multi- photon coherent destruction of tunneling and accounting for the suppression of the staircase jumps of d.c. current due to the memory effect. Specially, a novel blockade phenomenon is observed, showing distinctive oscillations in the field-induced current in the large bias voltage limit