Coherent control is a quantum mechanical based method for controlling dynamical processes by light. The basic principle is to control quantum interference phenomena typically by shaping the phase of a laser pulses  . The basic ideas have proliferated finding vast application in spectroscopy mass spectra, quantum information processing, laser cooling, ultracold physics and more.
The initial idea was to control the outcome of chemical reactions. Two approaches were pursued: In the time domain a pump dump scheme where the control is the time delay between pulses   and in the frequency domain, interfering pathways controlled by one and three photons. The two basic methods eventually merged with the introduction of optimal control theory  .
Experimental realisations soon followed. In the time domain  and in the frequency domain. Two interlinked developments accelerated the field of coherent control: Experimentally it was the development of pulse shaping by a spatial light modulator  and its employment in coherent control. The second development was the idea of automatic feedback control  and its experimental realization . 
Coherent control aims to steer a quantum system from an initial state to a target state via an external field. For given initial and final (target) states the coherent control is termed state-to-state control. A generalisation is steering simultaneously an arbitrary set of initial pure states to an arbitrary set of final states, i.e. controlling a unitary transformation. Such an application sets the foundation for a quantum gate operation.
Controllability of a closed quantum system has been addressed by Tarn and Clark. Their theorem based in control theory states that for a finite dimensional closed quantum system, the system is completely controllable, i.e. an arbitrary unitary transformation of the system can be realized by an appropriate application of the controls, if the control operators and the unperturbed Hamiltonian generate the Lie algebra of all Hermitian operators. Complete controllability implies state-to-state controllability.
The computational task of finding a control field for a particular state to state transformation is difficult and becomes more difficult with the increase in the size of the system. This task is in the class of hard inversion problems of high computational complexity. The algorithmic task of finding the field that generates a unitary transformation scales factorially more difficult with the size of the system. This is because a larger number of state-to-state control fields have to be found without interfering with the other control fields.
Once constraints are imposed controllability can be degraded. For example, what is the minimum time required to achieve a control objective . this is termed quantum speed limit.
Constructive approach to coherent control
The constructive approach uses a set of predetermined control fields for which the control outcome can be inferred. The pump dump scheme  in the time domain and the three vs one photon interference scheme in the frequency domain  are prime examples. Another constructive approach is based on adiabtic ideas. The most well studied method is Stimulated raman adiabatic passage STIRAP  which employs an auxiliary state to achieve complete state-to-state population transfer.
Optimal control as applied in coherent control seeks the optimal control field for steering a quantum system to its objective. For state-to-state control the objective is defined as the maximum overlap at the final time T with the state :
where the initial state is . The time dependent control Hamiltonian has the typical form:
where is a wavefunction like Lagrange multiplier and the parameter regulates the integral intensity. Variation of with respect to and leads to two coupled Schrödinger equations A forward equation for with initial condition and a backward equation for the Lagrange multiplier with final condition . Finding a solution requires an iterative approach. Different algorithms have been applied for obtaining the control field such as the Krotov method.
As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. 
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