Quantum control theory and applications: A survey <s> Bilinear models <s> The basic concepts of quantum mechanics Energy and momentum Schrodinger's equation Angular momentum Perturbation theory Spin The identity of particles The atom The theory of symmetry Polyatomic molecules Motion in a magnetic field Nuclear structure Elastic collisions Mathematical appendices. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Bilinear models <s> Control of quantum phenomena has grown from a dream to a burgeoning field encompassing wide-ranging experimental and theoretical activities. Theoretical research in this area primarily concerns identification of the principles for controlling quantum phenomena, the exploration of new experimental applications and the development of associated operational algorithms to guide such experiments. Recent experiments with adaptive feedback control span many applications including selective excitation, wave packet engineering and control in the presence of complex environments. Practical procedures are also being developed to execute real-time feedback control considering the resultant back action on the quantum system. This focus issue includes papers covering many of the latest advances in the field. Focus on Quantum Control Contents Control of quantum phenomena: past, present and future Constantin Brif, Raj Chakrabarti and Herschel Rabitz Biologically inspired molecular machines driven by light. Optimal control of a unidirectional rotor Guillermo Perez-Hernandez, Adam Pelzer, Leticia Gonzalez and Tamar Seideman Simulating quantum search algorithm using vibronic states of I2 manipulated by optimally designed gate pulses Yukiyoshi Ohtsuki Efficient coherent control by sequences of pulses of finite duration Gotz S Uhrig and Stefano Pasini Control by decoherence: weak field control of an excited state objective Gil Katz, Mark A Ratner and Ronnie Kosloff Multi-qubit compensation sequences Y Tomita, J T Merrill and K R Brown Environment-invariant measure of distance between evolutions of an open quantum system Matthew D Grace, Jason Dominy, Robert L Kosut, Constantin Brif and Herschel Rabitz Simplified quantum process tomography M P A Branderhorst, J Nunn, I A Walmsley and R L Kosut Achieving 'perfect' molecular discrimination via coherent control and stimulated emission Stephen D Clow, Uvo C Holscher and Thomas C Weinacht A convenient method to simulate and visually represent two-photon power spectra of arbitrarily and adaptively shaped broadband laser pulses M A Montgomery and N H Damrauer Accurate and efficient implementation of the von Neumann representation for laser pulses with discrete and finite spectra Frank Dimler, Susanne Fechner, Alexander Rodenberg, Tobias Brixner and David J Tannor Coherent strong-field control of multiple states by a single chirped femtosecond laser pulse M Krug, T Bayer, M Wollenhaupt, C Sarpe-Tudoran, T Baumert, S S Ivanov and N V Vitanov Quantum-state measurement of ionic Rydberg wavepackets X Zhang and R R Jones On the paradigm of coherent control: the phase-dependent light-matter interaction in the shaping window Tiago Buckup, Jurgen Hauer and Marcus Motzkus Use of the spatial phase of a focused laser beam to yield mechanistic information about photo-induced chemical reactions V J Barge, Z Hu and R J Gordon Coherent control of multiple vibrational excitations for optimal detection S D McGrane, R J Scharff, M Greenfield and D S Moore Mode selectivity with polarization shaping in the mid-IR David B Strasfeld, Chris T Middleton and Martin T Zanni Laser-guided relativistic quantum dynamics Chengpu Liu, Markus C Kohler, Karen Z Hatsagortsyan, Carsten Muller and Christoph H Keitel Continuous quantum error correction as classical hybrid control Hideo Mabuchi Quantum filter reduction for measurement-feedback control via unsupervised manifold learning Anne E B Nielsen, Asa S Hopkins and Hideo Mabuchi Control of the temporal profile of the local electromagnetic field near metallic nanostructures Ilya Grigorenko and Anatoly Efimov Laser-assisted molecular orientation in gaseous media: new possibilities and applications Dmitry V Zhdanov and Victor N Zadkov Optimization of laser field-free orientation of a state-selected NO molecular sample Arnaud Rouzee, Arjan Gijsbertsen, Omair Ghafur, Ofer M Shir, Thomas Back, Steven Stolte and Marc J J Vrakking Controlling the sense of molecular rotation Sharly Fleischer, Yuri Khodorkovsky, Yehiam Prior and Ilya Sh Averbukh Optimal control of interacting particles: a multi-configuration time-dependent Hartree-Fock approach Michael Mundt and David J Tannor Exact quantum dissipative dynamics under external time-dependent driving fields Jian Xu, Rui-Xue Xu and Yi Jing Yan Pulse trains in molecular dynamics and coherent spectroscopy: a theoretical study J Voll and R de Vivie-Riedle Quantum control of electron localization in molecules driven by trains of half-cycle pulses Emil Persson, Joachim Burgdorfer and Stefanie Grafe Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling Jean-Michel Coron, Andreea Grigoriu, Catalin Lefter and Gabriel Turinici Sliding mode control of quantum systems Daoyi Dong and Ian R Petersen Implementation of fault-tolerant quantum logic gates via optimal control R Nigmatullin and S G Schirmer Generalized filtering of laser fields in optimal control theory: application to symmetry filtering of quantum gate operations Markus Schroder and Alex Brown <s> BIB002
The state |ψ(t) of a closed quantum system evolves according to the Schrödinger equation BIB001 i ∂ ∂t where H 0 is the free Hamiltonian of the system (i.e., a Hermitian operator on H), is the reduced Planck's constant (we assume that = 1 in this paper), and the initial state has unit norm ψ 0 2 ≡ ψ 0 |ψ 0 = 1. For simplicity, we consider finite dimensional quantum systems, which are appropriate approximations in many practical situations. For an N -dimensional quantum system, H is an N -dimensional complex Hilbert space, and the eigenstates ) of H 0 form an orthogonal basis for H. In many situations, the control of the system may be realized by a set of control functions u k (t) ∈ R coupled to the system via time-independent Hermitian interaction Hamiltonians The goal in a typical quantum control problem defined on the system (5) is to find a final time T > 0 and a set of admissible controls u k (t) ∈ R which drives the system from the initial state |ψ 0 into a predefined target state |ψ f . The total Hamiltonian H(t) = H 0 + k u k (t)H k defines a unitary transformation (propagator) U (t) which can accomplish the transition from the pure state |ψ 0 to the pure state |ψ(t) ; i.e., Substituting (6) into (5), we can easily obtain According to the quantum state superposition principle  , the evolving state |ψ(t) can be expanded in terms of the eigenstates in the set D: Substituting (8) into (5), we let C(t) = {c 1 (t), c 2 (t), . . . , c N (t)} and obtain Equations (5), (7) or (9) are all referred to as finite dimensional bilinear models (BLM) of quantum control systems. If Equations (5), (7) and (9) are all controllable (see Section 3.1), conversion between them is easily carried out. For the BLM (7), we find a set of admissible controls to generate a desired unitary transformation U (t) and then calculate the desired state trajectory using  . For the BLM (9), we find a set of admissible controls to achieve desired coefficient trajectories C(t) and then obtain the desired state trajectory using BIB002 . Bilinear models are widely used to describe closed quantum control systems such as molecular systems in physical chemistry and spin systems in NMR. For example, consider a spin 1/2 system in a constant magnetic field along z-axis and controlled by magnetic fields along x-axis and y-axis  . We denote the Pauli matrices σ = (σ x , σ y , σ z ) as follows: The quantum control system under consideration can be described as follows: where the controls u