# .control

**..Introduction**

.Early developments of quantum control

The historical origins of quantum control lie in early attempts to use lasers for manipulation of chemical reactions, in particular, selective breaking of bonds in molecules. Lasers, with their tight frequency control and high intensity, were considered ideal for the role of molecular-scale `scissors’ to precisely cut an identied bond, without damage to others. In the 1960s, when the remarkable characteristics of lasers were initially realized, it was thought that transforming this dream into reality would be relatively simple. These hopes were based on intuitive, appealing logic. The procedure involved tuning the monochromatic laser radiation to the characteristic frequency of a particular chemical bond in a molecule. It was suggested that the energy of the laser would naturally be absorbed in a selective way, causing excitation and, ultimately, breakage of the targeted bond. Numerous attempts were made in the 1970s to implement this idea. However, it was soon realized that intramolecular vibrational redistribution of the deposited energy rapidly dissipates the initial local excitation and thus generally prevents selective bond breaking [27, 28, 29]. This process eectively increases the rovibrational temperature in the molecule in the same manner as incoherent heating does, often resulting in breakage of the weakest bond(s), which is usually not the target of interest.

.Control via two-pathway quantum interference

Several important steps towards modern quantum control were made in the late 1980s. Brumer and Shapiro [30, 31, 32, 33] identied the role of quantum interference in optical control of molecular systems. They proposed to use two monochromatic laser beams with commensurate frequencies and tunable intensities and phases for creating quantum interference between two reaction pathways. The theoretical

analysis showed that by tuning the phase difference between the two laser fields it would be possible to control branching ratios of molecular reactions [41, 42, 43]. The method of two-pathway quantum interference can be also used for controlling population transfer between bound states [44, 45] (in this case, the number of photons absorbed along two pathways often must be either all even or all odd to ensure that the wave functions excited by the two lasers have the same parity; most commonly,

one and three-photon ex-citations were considered).

The principle of coherent control via two-pathway quantum interference was demonstrated during the 1990s in a number of experiments, including control of population transfer in bound-to-bound transitions in atoms and molecules, control of energy and angular distributions of photo-ionized electrons and photo-dissociation products in bound-to-continuum transitions, control of cross-sections of photo-chemical reactions, and control of photo-currents in semiconductors. However, practical applications of this method are limited by a number of factors. In particular, it is quite difficult in practice to match excitation rates along the two pathways, either because one of the absorption cross-sections is very small or because other competing processes intervene. Another practical limitation, characteristic of experiments in optically dense media, is undesirable phase and amplitude locking of the two laser fields. Due to these factors and other technical issues (e.g., imperfect focusing and alignment of the two laser beams), modulation depths achieved in two-pathway interference experiments were modest: typically, about 25{50% for control of population transfer between bound states (the highest reported value was about 75% in one experiment), and about 15{25% for control of dissociation and ionization branching ratios in molecules. Two-pathway interference control is a nascent form of full multi-pathway control offered by operating with broad-bandwidth optimally shaped pulses.

.Pump-dump control

.Control via stimulated Raman adiabatic passage

.Control via wave-packet interferometry

.Quantum optimal control theory

.Control with linearly chirped pulses

.Control via non-resonant dynamic Stark effect

.Control of nuclear spins with radio frequency fields

**..Molecular Interactions: Light as controller**

.Molecular dipole interaction

.Representation of the Electric field

.Pictures in Quantum Mechanics

.Time-dependent Perturbation theory

.Quantum interference between pathways

**..(Classical)Optimal Control theory**

.Euler-Lagrange equations

.Examples of various types of cost functionals

.Linear and Bi-linear control systems

.The Pontryagin Maximum Principle

.Optimality conditions: Linear Control problems

.Analytic Solutions: General Guidelines

.Linear system: An example

**..Quantum optimal control theory**

.Introduction

.State manifolds and tangent spaces

.Controlled quantum mechanical systems

.Quantum optimal control theory

.Controllability of closed quantum systems

.Theoretical formulation of quantum optimal control theory

.Searching for optimal controls

.Applications of QOCT

.Open quantum systems

.Applications of QOCT for open quantum systems

**..Quantum control landscapes**

.Introduction

.Optimality of control solutions

.Pareto optimality for multi-objective control

.Landscape exploration via homotopy trajectory control

.Practical importance of control landscape analysis

.Experimental observation of quantum control landscapes