Some Topics in Modern Quantum Control

Control of atomic and molecular scale systems with quantum dynamics attracts nowadays high interest due to rich mathematical theory and various existing and prospective applications in physics, chemistry, and molecular biology including laser-assisted control of chemical reactions, quantum metrology, quantum optics, etc. Modern quantum technologies which might revolutionize our society like semiconductor revolution did in the second half of the twentieth century, are based on methods of quantum control [1–4].
Mathematical formulation of a quantum control problem included description of state space of the system, the dynamical equation, and specification of the target objective functional. The dynamics of the controlled quantum system is governed either by Schrödinger equation if the system is closed, that is, isolated from the environment, or by a master-equation if the system is open, that is, interacts with an environment. In both cases the evolution equation includes the control function which can be shaped laser field, spectral density of incoherent photons, or other external action. Objective functional can describe probability of transition from one state to another, average value of quantum observable, gate generation, etc. The goal of the optimal control is to find such a control function which maximizes the objective functional.
In this talk we will discuss recent progress in two very important and interesting topics in modern quantum control—controllability of open quantum systems and the analysis of quantum control landscapes.
Controllability of quantum systems deals with finding methods for transferring arbitrary initial states into arbitrary final states with admissible controls. We will discuss a method for a controlled engineering of arbitrary quantum states (density matrices) of $n$-level quantum systems which might be used for prospective quantum computing with mixed states [5].
Analysis of the control landscape, that is, graph of the objective functional, deals with the analysis of local but not global extrema (traps) of the objective functional. We will discuss the recent discovery of absence of traps for two-level systems [6,7] which are important as representing qubit—a basis building block for quantum computation, and for systems with infinite-dimensional state space, namely, for transmission coefficient of a quantum particle on the line passing through one-dimensional potential whose shape is used as a control [5]. For the latter, we consider a quantum particle of energy $E$ moving from the left in one dimensional potential $V(x)$ which is assumed to have compact support. Probability for the particle to appear far away on the right of the potential is the transmission coefficient $T_E[V]$. The transmission coefficient is a functional of the potential $V(x)$ and can be controlled by varying its shape. We show that the only extrema of the transmission coefficient as a functional of the potential $V$ are global maxima corresponding to full transmission [8]. This result is of high mathematical importance as the first result about absence of traps for quantum systems with infinite dimensional state space and of high practical significance as it says that manipulating by transmission coefficient is trap free.